References (Assessment 1)

References

Biggs, E. & Sutton, J. (1983). Teaching Mathematics 5 to 9: A classroom guide. London: McGraw-Hill Book Company (UK) Limited.

Bobis, J., Mulligan, J., and Lowrie, T. (2008). Chapter 9: Promoting Number Sense: Beyond Computation in Bobis, J.; Mulligan, J.; and Lowrie, T, Mathematics for children: challenging children to think mathematically, Frenchs Forest, NSW: Pearson Education Australia, pp.215-242. Retrieved from Queensland University of Technology Course Materials Database.

Clements, D.H. (1999). Subitizing: what is it? Why teach it? Teaching Children Mathematics, 5 (7), 400-405. Retrieved from Queensland University of Technology Course Materials Database.

Copley, J. V. (2010). The Young Child and Mathematics (2^nd Ed.). United States of America: National Assoiation for the Education of Young Children.

Irons, C. (2003). The Animal Parade: A fold-out counting book. Australia: Shortland Mimosa Publications.

Irons, R.R. (1999). Numeracy in early childhood, Educating Young Children: Learning and Teaching in the Early Childhood Years, 5(3), 26-32. Retrieved from Queensland University of Technology Course Materials Database.

Irons, R. R. (2003). Growing with Mathematics: Student Book. Chicago: Wright Group/McGraw-Hill.

Simmons, N. (2006). Mathematics in Early childhood: Exploring the issue. ACE Papers .

Yelland, N., Butler, D., Diezmann, C. (1999). Early Mathematical Explorations. United States of America: Pearson Publishing Solutions.

References for images

[Untitled photograph of Mathematics for children]. Retrieved September 5, 2011, from: http://cms.uca.edu:8080/acmse/equals/

[Untitled photograph of Mathematics for children]. Retrieved September 5, 2011, from: http://www.coolmathgames.ca/youchildteacher.html

[Untitled photograph of Mathematics]. Retrieved September 5, 2011, from: http://estherlke.blogspot.com/2010/08/people-differ.html

Reference for video

DavidOsaka. (2008, April 3). Counting numbers 1-10. Retrieved from http://www.youtube.com/watch?v=F5QLp9Wxrrg

When I was young...

During early years of primary school, my teacher encouraged us to use count on strategies. We used our fingers to do the simple addition and subtraction. However, when it comes to bigger number (more than 10), my teacher taught us to use the number lines (1-20) and empty number lines (up to 100). These methods are more effective and efficient because the methods are easier, simple and we can get the correct answers immediately. Furthermore, my teacher also provided us with many Mathematics exercises and quizzes every day. The exercises and quizzes are purposely to develop our thinking skills and mental computation skills (as we have to use number lines to find the answers). Other than that, my teacher also taught us the written strategies whenever the mental strategies are inadequate or some of my classmates did not really understand how to use mental computation. Therefore, I was using both mental and written computation during my primary school years and I realized that both methods are applicable and useful to solve any Mathematics problem.

On the other hand, my teacher taught us some different strategies when learning multiplication and division. Firstly, my teacher used direct modeling strategies when teaching multiplication. He brought several objects and modeled the multiplication process. However, after learning and understanding the basic concept of multiplication, we have to memorize the multiplication from multiple of 2 until 12. However, learning division was slightly different from learning multiplication where we didn’t have to memorize the division, but he taught us the sets or regroup strategies of division where he brought real objects to the classroom and demonstrated the division process.

In conclusion, I realized that using these mental and written computation strategies are very important as the basic knowledge of Mathematics in order for me to solve many complicated Mathematics problems in the future.

::Mental computation of addition and subtraction strategies::

Mental computation = "Mentally estimates and calculates addition and subtraction to 100 strategies based on ones and knowledge of number facts" (Bobis, Mulligan & Lowrie, 2008, p.220).

There are several models/representations of mental computations of addition and subtraction strategies which are:-

•  Tens frame
• Hundred charts
• Empty number lines

Tens frame

Two different ways of showing 8:-

Addition by using tens frame:- 7 + 5 

Fill in the tens frame with blue counters, thus 7 + 5 = 10 + 2 = 12

 Hundreds chart

"The hundreds chart is a well-known representation of numbers 1-100 (or 0-99) that has a multitude of uses for developing counting, patterning, base ten, addition and subtraction, and multiplication and division fact knowledge. The use of partitioning and sequencing ('split and jump') methods can be assisted through the complementary use of the hundreds chart" (Bobis, Mulligan & Lowrie, 2008, p.223).

Addition by using hundreds chart:-

57 + 24 = 81

(Put the purple counter at 57)
57 + 20= 77 (Put the orange counter at 77)  
77 + 4 = 81 (Move the orange counter to 81)

Empty number lines

Empty number lines is an effective representation for addition as the children can do the calculation by the 'split and jump' method in the number sequence.

'split and jump' method for addition

'split and jump' method subtraction

alternative 'split and jump' method for addition

alternative 'split and jump' method for subtraction

Teacher could use these models/representations in order to support children's mental computation development of addition and subtraction strategies effectively and efficiently.

::Number facts strategies for addition/subtraction::

There are several number facts strategies that children use for addition and subtraction which are:-

1. Count on from larger:
1. 8 + 48 = 48, 49, 50, 51, 52, ..... 56
2. 11 + 54 = 54, 55, 56, 57, 58, 59, 60, .....65
2.  Doubles: 
1. 9 + 9= 18 
2. 12+12 = 24
3. Near doubles: 
1. 12 + 13 = (12 + 12 + 1) = 25
2.  24 + 25 = (24 + 24 + 1) = 49
4. Halving:
1. 26 - 13 = 13
2. 16 - 8 = 8
5. Near halving: 
1. 27 - 13 = 14 = (26 - 13 = 13 + 1= 14)
2. 17 - 8 = 9 = (16 - 8 = 8 + 1 = 9)
6. Bridging to 10, 100 or nearest decade or multiple of 10:
1. 59 + 21 = 60 + 20 = 80
2. 89 + 91 = 90 + 90 = 180                               

::Early multiplication/division::

Multiplication

“Multiplication is: 1.An efficient way of adding sets of equal numbers of objects. 2. Magnification, often associated with enlargement or scale. The children will understand multiplication both as an efficient way of adding equal numbers and as magnification, and will be able to use the language patterns and to record multiplication in the accepted form” (Biggs and Sutton, 1983, p.52-53).

For example:
There are 3 cherries on a cupcake. How many cherries are there on 5 cupcakes?
5 x 3 = 15

This is the example of task sheet that the teacher could use to teach multiplication:

Taken from: Irons, R.R. (2003). Growing with Mathematics: Student Book. Chicago: Wright Group/McGraw-Hill.(page 89)

This task sheet requires the children to draw the legs of the people in the pictures. Then, they have to count the total number of legs in each group by multiplying the number of legs with the total number of people in each group. For example: 2 x 4 = 8 legs in all (first picture).

Division

“There are two aspects of division and three associated language patterns. The two aspects are sharing and subtraction” (Biggs and Sutton, 1983, p.62).

• Sharing

“In sharing, a collection of objects (or a quantity) is divided into a given number of equal sets (or parts); objects in a set are then counted. The two language patterns for sharing are:-

           # Share a number or marbles among 4 children. How many does each have?

            #Find one quarter of the set of marbles”

                                                                                                (Biggs and Sutton, 1983, p.62)

This is the example of task sheet that the teacher could use to teach about the concept of sharing in division process:

Taken from: Irons, R.R. (2003). Growing with Mathematics: Student Book. Chicago: Wright Group/McGraw-Hill.(page 91)

This task sheet requires the children to find the total amount of the objects, and then shared the total amount of the object evenly among 2 boxes. Finally, the children will find the total amount of the objects in each box. For example: 10 divided by 2 = 5 objects in each box.

• Subtraction

“The subtraction aspect is sometimes called the grouping or measuring aspect. Its language pattern is:

            #How many sets of a given number of objects can be made from this collection?”

                                                                                       (Biggs and Sutton, 1983, p.62)

For example: 
There are 21 apples in the basket. How many people will get the apples if each of them get 3 apples? 
21 divided by 3= 7 people

::Early addition/subtraction::

Addition

“Addition is the putting together of two or more sets of objects (or numbers) and finding the total. The concept of addition as combining two or more sets and finding the total will be acquired, and the accepted method of recording will be used” (Biggs and Sutton, 1983, p.35).

For examples:

There are 4 tennis balls in the box. Erinne puts 5 more tennis balls in the box. How many tennis balls are there in the box? 

4 + 5 = 9

These are the examples of task sheet that the teacher could use to encourage the children's learning about addition:

Taken from: Irons, R.R. (2003). Growing with Mathematics: Student Book. Chicago: Wright Group/McGraw-Hill.(page 48)

Taken from: Irons, R.R. (2003). Growing with Mathematics: Student Book. Chicago: Wright Group/McGraw-Hill.(page 69)

These task sheets require the children to count the amount of the objects in the pictures that will help them improving their addition skills.

Subtraction

“The concept of subtraction as:1. Taking away from one set, 2. Comparison of two sets, will be required, and the appropriate language patterns and accepted methods of recording will be used” (Biggs and Sutton, 1983, p.42).

• Taking away situation

“The most familiar of the three language patterns is that for the ‘take away’ situation in which the starting point is one set of objects from which objects are removed, used, or taken away. The language in this activity refers to taking away one set and having some objects left” (Biggs and Sutton, 1983, p.42).

For example:
There are 12 lollipops on the table. Joshua takes 3 lollipops away. How many lollipops left on the table?
12 - 3 = 9

• Comparison situation

“In these situations the starting point is two sets which remain unchanged throughout the operation. The comparison of the two sets (or two qualities) is the stage where subtraction is applied” (Biggs and Sutton, 1983, p.42).

For example: 
Amani has 6 marbles. Nani has 4 marbles fewer than Amani. How many marbles does Nani have?
6 - 4 = 2

This is the example of task sheet that might help the children to learn more about subtraction:

Taken from: Irons, R.R. (2003). Growing with Mathematics: Student Book. Chicago: Wright Group/McGraw-Hill.(page 87)

This task sheet requires the children to find the 'missing amount' by giving the total and the other amount as clues. This activity is quite challenging, yet it will provide opportunity to the children to improve their subtraction skills.

::Backward number sequence/number word before::

Backward number sequence 

• a process of having a number sequence in a backward order such as 10,9,8,7,6,5 and so on.

 Number word before

• a process to identify the missing number in a decreasing number sequence such as 5,4,3,_,1. The missing number is 2.

Teacher can encourage the children to play forward number sequence such as http://resources.oswego.org/games/spookyseq/spookycb2.html

::Forward number word sequence/number word after::

Forward number sequence

• a process of having a number sequence in a correct order such as 1,2,3,4,5 and so on.

Teacher can provide several activities to encourage children in learning and understanding number sequence.

For example:- 

Asks the children to line up in front of the classroom in forward number sequence.

Asks the children to hang the forward number sequence on a string.

Number word sequence

• a process to identify the missing number in an increasing sequence number such as 1,2, _ , 4, 5. The missing number is 3.

Teacher can encourage children to play the number sequence games. For example:- http://resources.oswego.org/games/spookyseq/spookyseq2.html

::Counting::

Counting is a basic process to identify the number of the objects in a group. Children will start learning to count from 1 to 10 and they will continue to count for bigger number. There are several ways of teaching children to count. The easiest way to teach is by using fingers. Teacher could also use other alternatives such as storybook or songs. 

For example:-

       Taken from: Irons, C. (2003). The Animal Parade: A fold-out counting book. Australia: Shortland Mimosa Publications.                                           

This book is colourful and interesting. As you can see, each of the page has the number at the top of the page, number word highlighted in the story, and pictures that symbolize the number. Thus, it will be easier for children to read while counting the pictures in each page for each number.

The other way is by using a counting song:-

http://www.youtube.com/watch?v=F5QLp9Wxrrg

Basically, both the storybook and song have the same concept of teaching and learning counting where both of them highlighted the number and use pictures to symbolize the number. This is a suitable and appropriate strategies to attract the children's interest to learn counting. 

These are the examples of task sheet that the teacher could give to the children as the reinforcement of counting skills. Besides reading the storybook and singing the song, these worksheets will also help the children to understand more about counting. 

Taken from: Irons, R.R. (2003). Growing with Mathematics: Student Book. Chicago: Wright Group/McGraw-Hill.(page 22)

Taken from: Irons, R.R. (2003). Growing with Mathematics: Student Book. Chicago: Wright Group/McGraw-Hill.(page 20)

Taken from: Irons, R.R. (2003). Growing with Mathematics: Student Book. Chicago: Wright Group/McGraw-Hill.(page 19)

::Subitizing::

“From a Latin word meaning suddenly, subitizing is the direct perceptual apprehension of the numerosity of a group” (Clements, 1999, p.400).

Subitizing is a process of recognizing the numerosity of a group quickly; looking at the quantity for a short time and then being able to tell how many are in the groups without counting each object (Copley, 2010).

Teacher can encourage children with subitizing by doing some activities such as:-

Using tens frame flash card with dots (the card shows 10)

The card shows 5

Using domino flash cards (the card shows 10)

 

These activities are vital among young learners because “subitizing is a fundamental skill in the development of students’ understanding of number” (Baroody, 1987, p.87 as cited in Clements, 1999, p.404). Subitizing also prepares children to pay attention and be alert with numbers especially when they are solving Mathematics problems in the future.

This is the example of task sheet that the teacher could use in the classroom during subitizing activity:

Taken from: Irons, R.R. (2003). Growing with Mathematics: Student Book. Chicago: Wright Group/McGraw-Hill.(page 55)

This task sheet has slightly the same concept as the dominoes flash cards, yet it is more interesting because it requires the children to colour the pictures that have a particular attribute (dominoes pictures that shows ten). Both activities provide the opportunity for the children to learn and improve their subitizing skills.

Workshop 2

After learning about the Beginning Processes, now we are moving on to new topics about early number sense and developing computation in the early years. The subtopics are: 

subitizing

counting

forward number word sequence/number word after

backward number word sequence/ number word before

early addition/subtraction

early multiplication/division

number facts strategies for either addition and subtraction or multiplication and division. 

Let us explore these topics.....

Oh Mathematics!

How do you view Mathematics?

When I was young, I think that Mathematics is all about numbers. But, when I grown up, I realize that Mathematics is not merely about numbers but it occurs in our daily life and routines such as time, money, space and measurement. Personally, I love Mathematics because it is fun, challenging and interesting. But, sometimes Mathematics can be 'scary' and makes me cry especially when I am lost and fail to find the problem solving. Nevertheless, my father, who is a former Mathematics teacher always encourage me to keep trying in finding the possible solutions and never give up as some Mathematics problems do have several methods and solutions.

How do you view Mathematics learning?

Mathematics learning at school is a process where the teacher helps the children to ‘realize’ Mathematics that happens around them. Learning Mathematics is not just drilling or chalk and talk approach but the teacher should be able to make the children understand mathematics by using appropriate strategies and activities. According to Clements (2001) “the most powerful mathematics for a preschooler is usually not acquired while sitting down in a group lesson but is brought forth by the teacher from the child’s own self-directed, intrinsically motivated activity” (p. 8, as cited in Simmons, 2006). Moreover, the teacher should be able to provide several interesting and interactive activities in order to make the mathematics learning meaningful and purposeful for the children. For example, the teacher could provide the students with objects that are familiar with their real-life such as building blocks, fruits, and toys and let the children learn by play. 

What are your assumptions about early childhood Mathematics?

Early childhood Mathematics is a process or situation of preparing the children with the basic concepts of Mathematics such as the Beginning Processes before the children learn Mathematics as a core subject at the school. The processes are very important to develop the children's creativity as well as other thinking skills.

::Patterning::

“Patterns are formed by the repetition of objects or pictures and are recognizable and predictable” (Irons, 1999, p.31).

“Patterning involves the repetition of a sequence of items or events” (Yelland, Butler & Diezmann, 1999, p. 12)

For example:

Patterning according to the colours of the koalas; red-yellow-blue-green and so on.

 Teacher should encourage the children to talk or describe about the patterns that they create during the patterning activities. This will improve the children's creativity and reasoning skills.  

These are the examples of task sheet that the teacher could give to the children when teaching about patterning:

Taken from: Irons, R.R. (2003). Growing with Mathematics: Student Book. Chicago: Wright Group/McGraw-Hill. (page 30)

Taken from: Irons, R.R. (2003). Growing with Mathematics: Student Book. Chicago: Wright Group/McGraw-Hill. (page 53)

These task sheets help the children to identify and recognize the patterns in each group. The children will be able to draw the next pictures if they recognized the patterns.

 

::Ordering::

“The process of ordering involves arranging objects, pictures, groups or events according to the relations between them based on increasing or decreasing amounts of attributes” (Irons, 1999, p.30)

“Ordering is an extension of comparing and involves sequencing three or more items or events according to a specific attribute” (Yelland, Butler & Diezmann, 1999, p.11)

For example:

Ordering the size of round surfaces (From left to right)= small - smaller - smallest

Ordering the height of the blocks (From left to right) = tall - taller - tallest

Children will be able to do the ordering activity if they can detect and identify attributes, making comparisons and detect differences (Irons, 1999). Also, the teacher should encourage the children to use “-est” words such as tall-taller-tallest/ big-bigger-biggest to describe the attributes of the objects.